Beating Blackjack

Beating the 21 + 3 Blackjack Side Bet: Some Really Cool Graphs

February 7, 2014 – 08:19 am

It continues to amaze me that my posts on the 21 + 3 blackjack side bet are among the most frequently viewed articles on this site. This post (Beating the 21+3 Blackjack Side Bet) tops the list. Because today is Wednesday, I created a computer program to look at how the edge for 21 + 3 changes as hands are dealt from a shoe. To be more specific, I considered a six-deck shoe with the cut card placed at 260 cards. For each hand, my program computed the exact edge for the 21 + 3 side bet based on the exact composition of the remaining cards in the shoe. I then used these edges on each hand to create a graph for the house edge for the 21 + 3 blackjack side bet over that shoe. I then repeated this experiment for 30 shoes.

As a reminder, the 21 + 3 blackjack side bet is based on the values of the three cards consisting of the player’s first two cards and the dealer’s up-card. If these three cards form a straight, a flush or trips, then the player wins 9-to-1. Otherwise the player loses. When I started this blog, I didn’t have a plan for how I was going to present my analysis. In particular, looking back at my original post about 21 + 3, I didn’t even give the combinatorial analysis. The following table corrects this omission:

The house edge is 3.2386%

The hit frequency is 9.6761%

The standard deviation is 2.9563

As for the computer simulations, to make the data a bit easier to compare and contrast, I plotted 10 shoes per graph and created three graphs (see below). Each line in each graph represents the house edge for the 21 + 3 blackjack side bet over a single six-deck shoe with the cut card placed at 260 cards (1 deck from the end). Click on an image to see it full-sized.

Everything starts at the same value at the far left of each graph, -0.032386 (the house edge). As each of these 30 shoes proceeds, their edges separate (become more “wiggly”) and move through a progressively wider range of values. These graphs illustrate exactly why deck penetration is so crucial to advantage play. Things don’t always favor the player, but the deeper the penetration, the more chance there is of getting an opportunity. For example, in the first graph above, there is an opportunity to make a 21 + 3 bet with more than a 25% edge over the house!